Show that f is bounded.

Suppose f: K-> (- , ), K is compact, and f has a finite limit at each point of K, but may not be continuous on K. Show that f is bounded by using the definition of compactness in terms of open covers.

I think this is the def. I'm supposed to be using. A set E is compact iff, for every family { } for some alpha in A of open sets such that EC alpha in A G_alpha, there is a finite set {alpha1,...,alphan} C A such that ECUi=1 to infinity G_(alphai).

Also, how would we show it if we use the sequential characterization of compactness?

Suppose f: K-> (- , ), K is compact, and f has a finite limit at each point of K, but may not be continuous on K. Show that f is bounded by using the definition of compactness in terms of open covers.

I think this is the def. I'm supposed to be using. A set E is compact iff, for every family { } for some alpha in A of open sets such that EC alpha in A G_alpha, there is a finite set {alpha1,...,alphan} C A such that ECUi=1 to infinity G_(alphai).

Also, how would we show it if we use the sequential characterization of compactness?